Phase mixing and the Vlasov equation in cosmology

TL;DR

This paper analyzes the decay and phase mixing effects of the Vlasov equation in cosmological models with different expansion rates, revealing enhanced decay rates and phase mixing phenomena depending on initial data regularity and expansion rate.

Contribution

It introduces a detailed analysis of phase mixing and decay rates for the Vlasov equation on expanding cosmological backgrounds, including the critical radiation-filled universe case.

Findings

Spatial density decays as t^{-6q} for expansion rate t^q with 0<q<1/2

Enhanced decay rates due to phase mixing depend on initial data regularity

Logarithmic and super-logarithmic enhancements occur at the radiation case q=1/2

Abstract

We consider the Vlasov equation on slowly expanding isotropic homogeneous tori, described by the Friedmann--Lema\^itre--Robertson--Walker cosmological spacetimes. For expansion rate $t^q$, with $0< q<\frac{1}{2}$ (excluding certain exceptional values), we show that the spatial density decays at the rate $t^{-6q}$ and that, when the spatial average is removed, the density decays at an enhanced rate due to a phase mixing effect. This enhancement is polynomial for Sobolev initial data and super-polynomial, but sub-exponential, for real analytic initial data. We further show that, when the expansion rate is the borderline $t^{\frac{1}{2}}$ -- the rate which describes a radiation filled universe -- a degenerate phase mixing effect results in a logarithmic enhancement for Sobolev initial data and a super-logarithmic enhancement (in fact, a gain of $\exp(-\mu(\log t)^{\epsilon})$ for some $\mu,\epsilon>0$) for analytic initial data. The proof is based on a collection of commuting vector fields, and certain combinatorial properties of an associated collection of differential operators. The vector fields are not explicit, but are shown to have good properties when $t$ is large with respect to the momentum support of the solution. A physical space dyadic localisation is employed to treat non-compactly supported (in particular, non-trivial real analytic) but suitably decaying solutions.